and numerical experiments, ensuring the accuracy of our dVi S½ = -mp (7) i simulation methodology. Size effects at a given mass load- dt ing using solid particles are examined and compared to SE = S½i · Vi. (8) analytical results for an equivalent gas in which the driven multiphase flow regime is reformulated as a dense gaseous where V is the dispersed phase velocity, mp is the mass in environment. A droplet breakup model is then introduced the dispersed phase and A p represents the mass added to and the effects of breakup on the passing front are ex- the system if the dispersed phase vaporizes when heated; amined. As the shock passes through the seeded region, for solid particles SÁ = 0. The momentum transfer from high temperature and pressure conditions become preva- the dispersed phase to the gas phase is given by S½ and, lent. When employing liquid fuels, this exposure results in neglecting heat transfer between phases, SE represents the droplet heating and burning. We thus investigate the ef- energy transfer between phases. fects of energy release from the burning fuel droplets which The radial direction is divided into 50 evenly sized trail the shock as it propagates downstream in an effort to computational cells and an adaptive 6000 cell grid is used determine the conditions necessary for the formation of a in the axial direction. The axial location of the pressure coupled detonation wave. front is tracked and the grid spacing is halved in the area immediately in front of and behind the shock. Away from the shock, an evenly spaced grid is used. 2. Governing equations and numerical model The basic shock tube geometry used is shown in Fig. 1. An 2.1. Particle motion axisymmetric geometry is assumed and consists of a closed cylindrical tube with a gaseous high pressure region at The driven section of the shock tube (P1) is seeded with pressure P4 which drives the lower pressure (P1) section. a monodisperse collection of stationary spherical particles The governing equations of motion used to model the gas or fuel droplets. The initial seeding is performed in such phase are a manner that the mass loading remains constant both "Á radially and axially throughout the seeded region. A small = -" · (ÁU) +SÁ (1) "t (10 cm) region at P1 is left unseeded in order to allow the shock wave to develop before entering the seeded region "(ÁU) = -" · (ÁUU) -"p + S½ (2) of the tube. The system is considered dilute in nature so "t that we may ignore any particle-particle interactions. "E A Lagrangian approach is used to track particles in the = -" · (EU) -"· pU + SE (3) "t gas phase (Squires and Eaton 1990). Under the assump- where tion that the density of particle Áp is much larger than the 1 density of the surrounding gas Ág the particle equation of E = e + ÁU2 (4) 2 motion reduces to e = p/(Å‚ - 1); (5) dVi(t) [Ui(Y, t) - Vi(t)]f(Rep) = (9) Á, p, e, U, and Å‚ are the density, pressure, internal en- dt Äp ergy, velocity, and specific heat ratio. These conservation equations for mass, momentum and energy are solved us- dYi(t) = Vi(t) , (10) ing the Flux-Corrected Transport (FCT) algorithm (Boris dt E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets 335 1.5 where V(t) and Y(t) are the velocity and position of the particle and U is the local gas phase velocity. The Stokes 1.45 response time Äp is defined as Current Sommerfeld, Exp. (1985) Ápd2 Sivier, et al., Num. (1994) p 1.4 Äp = , (11) 18µ 1.35 where dp is the particle diameter and µ the dynamic viscosity of the surrounding gas. Gravity effects are ne- 1.3 glected. The coefficient f(Rep) is a scalar function of the particle Reynolds number Re = |U - V|dp/½air. A non- linear drag law (Clift et al. 1978) is used with f(Re) = 1.25 1.0+0.15Re0.687. We note that other drag laws have been suggested as alternatives (e.g. Sommerfeld 1985; Sivier 1.2 0 100 200 300 et al. 1994; Henderson 1976) under high velocity condi- z, cm tions. Our experiences though (Chang et al. 1995) have Fig. 2. Shock position vs shock Mach number at M0 =1.49, shown that employing different drag laws results in quali- · =0.63, and dp0 =27µm tatively similar results with only small quantitative differ- ences present, especially when particle sizes and slip veloc- for each simulation, we can examine effects of particle size ities are small. A second order predictor-corrector method explicitly. Because many engineering applications involve is used to integrate the particle equations of motion and liquid fuel injection, a droplet breakup model is introduced determine particle or droplet velocity and position. In a one-way coupled system the particle mass load- and the effects of breakup on the attenuation rate of the shock are examined. Finally, results for an energy release ing is assumed to be small enough that any effects the model in which energy from droplet burning is added to presence of the dispersed phase have on the gas-phase can be neglected. Thus, the local gas phase velocity has a di- the system are presented. rect bearing on particle motion while the reverse is not true. However, in the cases examined here, the particle mass loading is high enough that return effects cannot 3.1. Comparison with previous studies be completely ignored and a two-way coupled system is employed. To account for the return effects from the dis- For the unseeded flow, results for simulations using a pres- persed particles to the gas-phase the particle momentum sure ratio of P4/P1 = 4 agree with the theoretical re- and energy are calculated for each particle location. A lin- sults for the basic bursting diaphragm problem. The cal- ear weighting scheme based on cell volume is used to find culated shock speed is M0 =1.49 with a shock strength the corresponding source terms (SÁ, S½ and SE) for the of P2/P1 =2.43. In order to validate our multiphase flow gas-phase calculation at the surrounding grid locations. simulations, we compare our result with previous exper- In order to achieve high mass loadings without the nec- imental and numerical works. Figure 2 shows the Mach essary computational expense of tracking every particle in number variation of the shock wave, where the Mach num- the system, a virtual particle method is employed. Each ber Ms is defined as the shock speed divided by the speed particle in the simulation acts as a marker or carrier for a of sound in the gas at P1 in the absence of particles; group of virtual particles with a center of mass located at the simulation particle position. It is assumed that each of Ms = cs/(Å‚P1/Ág)1/2. (12) the virtual particles has the same size and mass of the sim- ulation particle but their velocity and location are not ex- The dispersed region consists of 27 µm diameter glass plicitly calculated as they are assumed to move with their beads of density Áp=2.5 g/cm3 suspended in air at pres- associated carrier particle velocity and position. However, sure P1=1 atm. The mass loading, given as the ratio of the the momentum and energy from the virtual particles are mass of the dispersed phase to the mass of the gas phase included in the coupling feedback source terms and are in the driven section, is · =0.63. Results from the current assumed to be located at the same position as the carrier simulation are shown to be in good agreement with those particle. For these simulations 100,000 real particle are from experiments by Sommerfeld (1985) and numerical tracked, with the number of virtual particles employed simulations by Sivier et al. (1994). appropriately chosen in order to achieve the desired mass loading. Simulations were performed on a Cray C90 vector computer and an SGI Origin 2000 parallel computer. 3.2. Solid particles at · =0.50 In order to determine the effect of particle size, a series of monodisperse simulations were performed with particle 3. Results and discussion sizes ranging from 12.5 µm d" dp d" 100 µm. The particle First, we present results for monodisperse solid particles density was chosen as Áp = 1 g/cm3 and the viscosity of at constant mass loading. By using a different particle size the gas phase assumed to be equal to that of air at the local s M 336 E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets 4.6 140000 4.5 a) 120000 4.4 12.5µm 25µm 100000 50µm 4.3 100µm 4.2 80000 4.1 60000 4 3.9 40000 12.5µm 3.8 25µm 50µm 20000 100µm 3.7 3.6 0 0 100 200 300 -200 -100 0 100 200 z, cm z, cm 4E+07 Fig. 3. Shock position vs shock Mach number. Solid particles b) at · =0.50 and M0 =4.52 3E+07 (to the particle) pressure and temperature. The pressure ratio between the driver and driven sections of the tube was chosen as P4/P1 = 50 which gives rise, in an unseeded flow, to a shock with a strength of P2/P1 = 23.7 which 2E+07 travels downstream at a Mach number of M0 =4.52. While attenuation, or deceleration, of the shock front is expected for all the multiphase flow cases examined, 12.5µm 1E+07 25µm it is unclear whether gains in reduced drag from the use 50µm of smaller particles will be offset by the increased num- 100µm ber of particles used in order to maintain the same mass loading. Figure 3 shows the shock speed at different loca- -200 -100 0 100 200 tions in the shock tube. As the shock travels downstream, z, cm the exponential-like decay of its velocity can be seen. The Fig. 4a,b. Solid particles at t =0.9 ms, · =0.50 and M0 = shock speed is seen to decrease more rapidly with decreas- 4.52; a velocity distribution and b pressure distribution ing particle size and, for the 12.5 µm case, an equilibrium value is almost immediately attained. Note too that for all -1 "qj 1 "Ui cases examined it appears that the same equilibrium value + Äij . (14) (1 + º)Á "xj (1 + º)Á "xj is approached. For this length tube though, the shock in the 100 µm case does not reach a terminal velocity before If the equation of state is then written as the end of the tube is encountered. Simulations employing
a longer tube indicate that this same equilibrium velocity R is indeed attained. These results indicate that, while the p =(1 +º)Á T , (15) 1+º attenuation rate is increased when smaller particles are employed, there is no evidence that the increased number the system behaves as a perfect gas with density Áe = of particles leads to a lower terminal shock velocity. (1 + º)Á and the ratio of specific heats of the gas in this In fact, the final equilibrium value attained for a given modified system becomes mass loading may be obtained through closer examination of the full coupled equations of motion. By taking the limit Å‚(1 + ·´) of dp to zero and assuming a truly uniform distribution Å‚e = (16) 1+Å‚·´ of particles, the momentum equation may be written as (Marble 1970) where ´ = c/cp is the ratio of the specific heats of the par- ticles and the gas. These modified density and specific heat "Ui "Ui -1 "p 1 "Äij values may now be used to find the theoretical shock ve- + Uj = + , (13) "t "xj (1 + º)Á "xi (1 + º)Á "xj locity for this equivalent gas system from normal shock relations (Liepmann and Roshko 1957). The final shock where º = Áp/Ág and Äij is the deviatoric stress tensor. velocity attained in our multiphase flow simulations is ap- Similarly the energy equation may be written as proximately Me=3.73, which agrees with the analytical solution.
cv + ºc "T "T The axial variation of the gas phase velocity and pres- + Uj = sure in the tube at t=0.9ms are shown in Figs. 4a and 4b. 1+º "t "xj s M Velocity, cm/s 2 Pressure, dyne/cm E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets 337 2.65E+07 4.5 4.4 2.6E+07 4.3 solid tb=100µs 2.55E+07 4.2 tb=10µs tb=1µs 4.1 2.5E+07 4 12.5µm 25µm 2.45E+07 50µm 3.9 100µm 3.8 2.4E+07 0 0.0005 0.001 0.0015 0.002 0.0025 t, sec 3.7 0 100 200 300 Fig. 5. Time vs maximum pressure in the seeded region of the z, cm flow P2m. Solid particles at · =0.50 and M0 =4.52 Fig. 6. Shock position vs shock Mach number. Droplet breakup model at · =0.50 with dp0 =25 µm and M0 =4.52 Increased attenuation of the amplitude of the gas phase velocity is observed with decreasing particle size. We also note that the location of the leading edge of the shock de- creases with particle size indicating increased deceleration struct a detailed breakup model. To obtain some under- of the front for smaller particles. In the pressure distribu- standing of the effects of droplet breakup, a simple model tion, significant attenuation of the front can be seen at is assumed in this study. In this model, the droplet size the leading edge of the shock wave for large particles. The is assumed to be constant for a fixed amount of time tb, dip in pressure at the expansion front is due to the small corresponding to the breakup time, upon passage of the unseeded buffer region initially at pressure P1. For smaller shock. At tb it is assumed that some percentage of the particles, sharper leading and trailing edges are visible in- droplet turns immediately to vapor with the remaining dicating behavior approaching that of the equivalent dense mass replaced by a specified number of smaller droplets, gas. In fact, for the 12.5 µm diameter case, the pressure all of which are the same size. ratio across the shock front is the same as that for the Previous experimental studies (Ranger and Nicholls equivalent gas case with P2/P1=26.0. 1969; Yoshida and Takayama 1990; Pilch and Erdman Figure 5 shows the maximum pressure in the seeded 1987) indicate that under the conditions presented in these region P2m as a function of time. In the 12.5 µm case simulations, droplet breakup times are typically on the the pressure remains nearly constant, and closely approx- order of 100 µs for large droplets (dp = O(1000 µm)). imates that at the theoretical value for a gas. In the other Smaller droplets are expected to breakup more rapidly cases, the pressure is observed to increase with time with upon contact with the shock front. In the absence of the 25 µm and 50 µm cases clearly approaching the theo- more detailed information for the droplet size used here, retical limit. Again, while it appears that the 100 µm case a parametric study is conducted with a range of breakup will also reach this value, the shock reached the end of the times, 1 µs d" tb d" 100 µs. Upon breakup, each droplet tube before this could be confirmed. mass is converted into some vapor (mv) and 10 equally sized smaller droplets with the remaining mass. As be- fore, we start with an initial distribution of 25 µm diam- 3.3. Droplet breakup model eter droplets at 0.50 mass loading. Simulations were run in which the amount of droplet mass converted directly to When the dispersed phase consists of liquid droplets, ad- vapor upon breakup ranged from mv = 0 (no mass trans- ditional factors should be considered. Upon exposure to a fer) to mv = 100% (all of droplet converted to vapor). high velocity field such as that following a high Mach num- All other droplet and flow parameters remain unchanged ber shock wave, it is well known that droplets with Weber from the previous solid particle investigation. numbers greater than O(1) have a tendency to breakup Figure 6 shows the shock velocity at different axial lo- due to the high shearing forces experienced at the droplet cations using this breakup model with no droplet mass surface (Clift et al. 1978). The Weber numbers experi- transfer upon breakup (mv=0). Because the mass loading enced here, based on the surface tension values of liquid remains constant, the equilibrium value reached remains fuels typically found in propulsion applications, can range unchanged. In all cases, the shock is decelerated at a rate from 100 d" We d" 1000, depending on droplet size and faster than in the solid particle case with increasing at- slip velocity. Although several different breakup regimes, tenuation rates corresponding to smaller tb. Examination based on Weber Number, are known to exist (e.g. Jones of the axial pressure in the shock tube in these cases (not and Thomas 1992), sufficient information on the details of shown) shows little difference among the different cases, the breakup process is not available for the droplet sizes with a slight widening of the shock front observed as the and conditions of interest and hence it is difficult to con- breakup time increases. 2 s M 2m P , dyne/cm 338 E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets 4.5 4 4.4 3.5 mv=0 mv=25% 4.3 3 mv=50% mv=75% 4.2 mv=100% 2.5 4.1 2 4 1.5 3.9 1 3.8 0.5 3.7 0 0 10 20 30 40 50 0 100 200 300 z, cm z, cm Fig. 7. Shock position vs shock Mach number. Droplet Fig. 8. Mass loading at t =1.65 ms for solid particle simulation breakup model with vaporization at · = 0.50 with dp0 = at · =0.50 with dp0 =25 µm and M0 =4.52. The vertical line 25 µm, M0 =4.52, and mv=0, 25%, 50%, 75%, and 100% at z = 184 cm represents the location of the shock front The shock velocity for the droplet breakup model with simple d2-law, where the droplet diameter squared decays mass transfer is shown in Fig. 7 for a fixed tb =10 µs. Note linearly with time, provides a good approximation to the that only the interval 0 d" z d" 50cm is shown since the time dependent behavior of a liquid fuel in this environ- shock velocity remains relatively unchanged once the equi- ment. We therefore used the d2-law in this study with an librium value is reached. An increased attenuation rate is evaporation rate of ² =10-3 cm2/s. That is, observed when more of the initial droplet mass is allowed to turn directly into vapor (increasing mv). In all cases d(d2) p = -² (17) the equilibrium velocity reached is the same. This is to be dt expected as the long time behavior in the seeded flow is The energy release model employed is similar to that the same as in the equivalent gas and we may infer that used in our previous studies involving small amounts the combination of droplets and vapor will yield similar of dispersed high-energy fuels in a ramjet combustor results. (Chang and Kailasanath 1997, 1999). During the sim- ulation, droplet sizes and lifetimes are monitored and when a specified threshold is reached, droplets rapidly re- 3.4. Energy release model lease their energy. By specifying different thresholds, the For many propulsion applications the use of a gaseous fuel time until energy release may be controlled. This may be is impractical and the likely source of energy will come in thought of generically as effectively changing the induc- the form of a dispersed liquid phase in which droplets are tion time ti (time after passage of the front until energy heated and burn, releasing energy into the system. If the release begins) for energy release in a typical liquid hydro- energy release can couple with a passing shock wave, a carbon fuel droplet. For consistency the amount of energy detonation wave will be created, resulting in a very effi- released by each droplet, ep = 100 erg, is the same for all cient propulsion system (Kailasanath 1999). Thus, finding simulations regardless of droplet size at the time energy is the timescale over which energy is released which allows released. For larger amounts of energy release, more com- for such coupling is of high importance. Clearly, if energy plex models may be needed. As before, the initial droplet release occurs in a region close to the front, such coupling size is 25 µm, the mass loading is 0.50, and all other flow is possible. parameters remain unchanged. When considering the transitional case where droplets Examination of the mass loading in the shock tube start in a normal (subcritical) environment but finish at indicates that an area of very high mass loading exists near or super-critical environment, the work of Zhu and in the area directly behind the shock wave as droplets are Aggarwal (1999) indicates that a hydrocarbon fuel droplet carried with the traveling front. For example, Fig. 8 shows will most likely remain in a liquid state well past the crit- the axial variation of the mass loading at t =1.65 ms for ical temperature and pressure before changing to a va- the 25 µm solid particle case. The shock front is located at por state. Although detailed simulations were necessary z = 184 cm at this time. Of interest is whether this region to show this result, it suggests that a simpler model based of high mass loading is close enough to the leading front so on the burning or vaporization rate of the droplet may that energy released in this area can couple with the front. provide a suitable estimate of droplet size and lifetime. If so, this would allow some leeway in the induction time Examination of the burning rate found in such models as used. In order to determine the timescale in which effective well as results from experiments (Hsieh et al. 1991; Shuen energy release occurs, that is the timescale in which energy et al. 1992; Givlier and Abraham 1996) indicates that a release couples with the shock front, a series of simulations s M Mass Loading E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets 339 4E+07 4E+07 a) b) 3.5E+07 3.5E+07 3E+07 3E+07 2.5E+07 2.5E+07 2E+07 2E+07 1.5E+07 1.5E+07 1E+07 1E+07 5E+06 5E+06 0 0 -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 z, cm z, cm 4E+07 4E+07 c) d) 3.5E+07 3.5E+07 3E+07 3E+07 2.5E+07 2.5E+07 2E+07 2E+07 1.5E+07 1.5E+07 1E+07 1E+07 5E+06 5E+06 0 0 -300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 z, cm z, cm Fig. 9a d. Pressure distribution at t =1.65 ms. Energy release model at · =0.50 and ep = 100 erg/droplet with dp0 =25 µm, M0 =4.52 and droplet lifetimes of a ti = 1000 µs, b ti = 100 µs, c ti =10 µs, and d ti =1µs 4.8 sure distribution. While it is clear that in the ti = 1000 µs case energy release does not fully couple with the shock, 4.6 increased pressure due to energy release is seen to occur ti=1µs near the shock in the other cases indicating the possibil- ti=10µs ity of strong coupling. Examination of the shock velocities, ti=100µs 4.4 ti=1000µs as seen in Fig. 10, indicates that strong coupling, such as that seen in a detonation, occurs in the ti =1 µs and 10 µs 4.2 cases with little difference observed in both the accelera- tion rates and equilibrium velocities. The shock is imme- diately and rapidly accelerated to a velocity higher than 4 that in the unseeded case and this velocity is maintained as the front continues to travel downstream. In addition, 3.8 from this figure it is now apparent that only weak cou- pling occurs in the other cases. For example, while energy 3.6 release occurs immediately behind the shock front in the 0 100 200 300 z, cm ti =1 µs case, when ti = 100 µs energy release occurs in a region 7 9 cm behind the shock. This large spacing be- Fig. 10. Shock position vs shock velocity. Energy release tween the region of energy release and the shock front also model at · =0.50 and ep = 100 erg/droplet with dp0 =25µm accounts for the difference between the time energy release and M0 =4.52 begins and the time it is observed to accelerate the front; for ti = 100 µs energy release begins when the shock is were conducted using droplet lifetimes spanning several located at z = 5 cm but acceleration is not observed until orders of magnitude with 1 µs d" ti d" 1000 µs. z = 30 cm. Even then, attenuation due to the mass load- Figures 9a d show the axial pressure distribution at ing is seen to overcome the acceleration by energy release t =1.65 ms for four of the cases examined. The effects of with the shock attaining a final velocity still below that in energy release are seen as large oscillations in the pres- 2 2 Pressure, dyne/cm Pressure, dyne/cm 2 2 Pressure, dyne/cm Pressure, dyne/cm s M 340 E.J. Chang, K. Kailasanath: Shock wave interactions with particles and liquid fuel droplets the unseeded flow. Even for ti = 1000 µs case, the effects times. The amount of time that passes before energy re- of energy release can be seen in the abrupt jump in the lease begins is therefore large and energy release may fail shock velocity around 220 cm in Fig. 10. Consistently, the to couple with the shock wave leading to the absence of amount of increase in shock velocity is lower and occurs a detonation. When small droplets are used though, the later in this case. droplet heating time is small. Thus, energy release begins almost immediately upon passage of the front, increas- ing the likelihood of a successful detonation, especially at low to moderate seeding levels. Under certain conditions 4. Conclusions though, such as those observed by Kauffman et al. (1984), detonation failure may still arise. At very high mass load- When examining the results from these simulations of ings, the presence of small droplets can lead to an almost shock propagation in an environment heavily seeded with immediate deceleration of the shock and the shock may particles or droplets we can make several observations. not be strong enough to form a detonation. However, as The presence of a dispersed phase has a substantial effect long as the amount of energy released is high enough to on the behavior of the shock wave. For a specified mass overcome this deceleration, our results indicate that a sus- loading, it was found that the shock velocity was reduced tained detonation may still be achieved. in an exponentially decaying manner to the same velocity regardless of particle size. Increased attenuation rates were Acknowledgements. This work has been supported by the U.S. observed to correspond to smaller particle sizes. Interest- Office of Naval Research through the Mechanics & Energy ingly, the equilibrium velocity attained was the same as Conversion Division and the Naval Research Laboratory. A that predicted by the analytical results of Marble (1970) grant of HPC time from the DoD HPC Shared Resource Cen- obtained for the limit as the particle size goes to zero and ter, CEWES, is also gratefully acknowledged. The help of Dr. the fuel/gas mixture is replaced with an equivalent gas Gopal Patnaik in parallelizing the code is also gratefully ac- with a modified density and specific heat ratio. Although knowledged. the attenuation of the shock to this limiting value occurs over an axial distance, this result indicates that the ana- lytical solution can be used to predict the final velocity of References the shock in the seeded flow. 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